Calculus Examples

$y = 5 {(5 x^{2} + 5 x)}^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (5 {(5 x^{2} + 5 x)}^{3})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 {(5 x^{2} + 5 x)}^{3}$ with respect to $x$ is $5 \frac{d}{d x} [{(5 x^{2} + 5 x)}^{3}]$ .

$5 \frac{d}{d x} [{(5 x^{2} + 5 x)}^{3}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 5 x^{2} + 5 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $5 x^{2} + 5 x$ .

$5 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [5 x^{2} + 5 x])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$5 (3 u^{2} \frac{d}{d x} [5 x^{2} + 5 x])$

Step 3.2.3

Replace all occurrences of $u$ with $5 x^{2} + 5 x$ .

$5 (3 {(5 x^{2} + 5 x)}^{2} \frac{d}{d x} [5 x^{2} + 5 x])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $3$ by $5$ .

$15 ({(5 x^{2} + 5 x)}^{2} \frac{d}{d x} [5 x^{2} + 5 x])$

Step 3.3.2

By the Sum Rule, the derivative of $5 x^{2} + 5 x$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [5 x]$ .

$15 {(5 x^{2} + 5 x)}^{2} (\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [5 x])$

Step 3.3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$15 {(5 x^{2} + 5 x)}^{2} (5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x])$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$15 {(5 x^{2} + 5 x)}^{2} (5 (2 x) + \frac{d}{d x} [5 x])$

Step 3.3.5

Multiply $2$ by $5$ .

$15 {(5 x^{2} + 5 x)}^{2} (10 x + \frac{d}{d x} [5 x])$

Step 3.3.6

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$15 {(5 x^{2} + 5 x)}^{2} (10 x + 5 \frac{d}{d x} [x])$

Step 3.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$15 {(5 x^{2} + 5 x)}^{2} (10 x + 5 \cdot 1)$

Step 3.3.8

Multiply $5$ by $1$ .

$15 {(5 x^{2} + 5 x)}^{2} (10 x + 5)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 15 {(5 x^{2} + 5 x)}^{2} (10 x + 5)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 15 {(5 x^{2} + 5 x)}^{2} (10 x + 5)$